© Hindawi Publishing Corp. DUNKL WAVELETS AND APPLICATIONS TO INVERSION OF THE DUNKL INTERTWINING OPERATOR AND ITS DUAL

We define and study Dunkl wavelets and the corresponding Dunkl 
wavelets transforms, and we prove for these transforms Plancherel and reconstruction formulas. We give as application the inversion of the Dunkl intertwining operator and its dual.


Introduction.
We consider the differential-difference operator Λ α on R introduced by Dunkl [1] and called in the literature Dunkl operator on R of index (α +1/2) associated with the reflection group Z 2 , given by These operators are very important in mathematics and physics.They allow the development of generalized wavelets from generalized continuous classical wavelet analysis.Moreover, we have proved in [2] that the generalized two-scale equation associated with the Dunkl operator has a solution and then we can define continuous multiresolution analysis.
Dunkl has proved in [1] that there exists a unique isomorphism V α , called the Dunkl intertwining operator, from the space of polynomials on R of degree n onto itself, satisfying the transmutation relation Rösler has proved in [3] that for each x ∈ R there exists a probability measure µ x on R with support in the interval [−x, x], such that for all polynomials p on R, we have V α (p)(x) = R p(y)dµ x (y). (1.4) Next, Trimèche in [5] has extended the operator V α to an isomorphism from Ᏹ(R), the space of C ∞ -functions on R, onto itself satisfying the relations (1.1) and (1.2), and has shown that for each x ∈ R, there exists a unique distribution η x in Ᏹ (R), the space of distributions of compact support on R, with support in the interval He has shown also in [5] that the transposed operator t V α of the operator V α has the integral representation where ν y is a positive measure on R with support in the set {x ∈ R : |x| ≥ |y|} and f in Ᏸ(R), the space of C ∞ -functions on R with compact support, and t V α is an isomorphism from Ᏸ(R) onto itself, satisfying the relations and for each y ∈ R, there exists a unique distribution Z y in (R), the space of tempered distributions on R, such that In this paper, we are interested in Dunkl wavelets and associated Dunkl continuous wavelet transforms.More precisely, we give here a general construction allowing inverse formulas for the Dunkl intertwining operator and its dual.
The contents of this paper are as follows.In Section 2, we define and study the Dunkl intertwining operator and its dual.Section 3 is devoted to Dunkl wavelets and associated Dunkl wavelet transforms.In the last section, we give as application of the previous results inverse formulas for the Dunkl intertwining operator and its dual.
2. The Dunkl intertwining operator on R and its dual.We define and study in this section the Dunkl intertwining operator on R and its dual and we give their properties.Notation 2.1.We have the following notations: (i) D a (R) is the space of C ∞ -functions on R with support in the interval [−a, a]; (ii) S(R) is the space of C ∞ -functions on R, rapidly decreasing together with their derivatives; where (2.5) where where f e (resp., f 0 ) is the even (resp., odd) part of f , R α is the Riemann-Liouville integral operator defined in [5], and I is the operator given by (2.10) Let f be in Ᏹ(R) and g in D(R).The operator t V α , defined on D(R) by the relation where k α is the kernel given by relation (2.7).It is called the dual Dunkl intertwining operator.It has the following properties.
Theorem 2.4.For all f in D(R), where f e (resp., f 0 ) is the even (resp., odd) part of f , t R α is the Weyl integral operator defined in [4], and J is the operator given by (2.16) The inverse transform ( t V α ) −1 is given by where ( t R α ) −1 is the inverse operator of t R α .
(ii) The transform t V α satisfies the transmutation relation (2.18)

Classical continuous wavelets and Dunkl wavelets
3.1.Classical continuous wavelets on R. We say that a measurable function g on R is a classical continuous wavelet on R if it satisfies, for almost all x ∈ R, the condition where Ᏺ is the classical Fourier transform.Let a ∈]0, +∞[ and let g be a classical wavelet on R in L 2 (R).We consider the family g a,x , x ∈ R, of classical wavelets on R in L 2 (R) defined by where H a is the dilation operator given by the relation We define, for regular functions f on R, the classical wavelet transform T g on R by This transform can also be written in the form where * is the classical convolution product.The transform T g has been studied in [5].Several properties are given; in particular, if we consider a classical wavelet g on R in L 2 (R), we have the following results.
(i) Plancherel formula.For all f in L 2 (R), we have T g (f )(a, y)g a,x (y)dy da a , a.e., where for each x ∈ R, both the inner and the outer integrals are absolutely convergent, but possibly not the double integral.(i) σ (x,y,z), ρ(x, y, z), and W α (x,y,z) are the functions defined for all x, y, z ∈ R\{0} by where K α is the Bessel kernel; (ii) for all x, y ∈ R, µ α x,y is the measure on R given by where δ x is the Dirac measure; (iii) the Dunkl translation operators τ x , x ∈ R, are defined on Ᏹ(R) by (3.10) Definition 3.2.A Dunkl wavelet on R is a measurable function g on R satisfying, for almost all x ∈ R, the condition where and ψ α λ (z) is the Dunkl kernel given by The function g(x) = −(d/dt)α t (x) is a Dunkl wavelet on R in (R), and We deduce then these results from Theorem 2.4.Let a ∈]0, +∞[ and let g be a regular function on R. We consider the function g a defined by (3.17) It satisfies the following properties: (i) for g in L 2 α (R), the function g a belongs to L 2 α (R) and we have (ii) for g in (R) (resp., α • (R)), the function g a belongs to (R) (resp., α • (R)) and we have Let g be a Dunkl wavelet on R in L 2 α (R).We consider the family g a,x ,x ∈ R, of Dunkl wavelets on R in L 2 α (R) defined by where τ x , x ∈ R, are the Dunkl translation operators.Using (3.15), we deduce that the family g a,x , x ∈ R, given by is a family of Dunkl wavelets on R in (R).
Definition 3.5.Let g be a Dunkl wavelet on R in L 2 α (R).The Dunkl continuous wavelet transform S D g on R is defined for regular functions f on R by This transform can also be written in the form where * D is the Dunkl convolution product defined by 2 da a |y| 2α+1 dy. (3.26) But from Definition 3.2, we have for almost all y ∈ R, 2 y 2α+1 dy. (3.28) We then deduce the relation (3.25).
The following theorem gives an inversion formula for the transform S D g .
Theorem 3.7.Let g be a Dunkl wavelet on R in where for each x ∈ R, both the inner and the outer integrals are absolutely convergent, but possibly not the double integral.

Inversion of the Dunkl intertwining operator and of its dual by using Dunkl wavelets. Using the inversion formulas for the Dunkl continuous wavelet transform S D
g and classical continuous transform S g , we deduce relations which give the inverse operators of the Dunkl intertwining operator V α and of its dual t V α .Theorem 4.1.(i) Let g be a Dunkl wavelet on R in Ᏸ(R) (resp., (R)).Then for all f in the same space as g, where is the operator given by the relation Proof.We obtain these relations from Theorem 4.1 and the fact that t V α (g) 0 a = a −2α t V α (g) 0 a , D g a = a −2α D (g) a . (4.7) Theorem 4.3.Let g be a Dunkl wavelet on R in α • (R).Then, V α S ( t Vα(g)) (f )(a, •) (y)g a,x (y)|y| 2α+1 dy da a 2α+1 ; (4.8) (g) (f )(a, •) (y) t V α (g) a,x (y)dy da a 2α+1 . (4.9) Proof.We deduce (4.8) and (4.9) from Theorems 4.2, 2.4 and relation (2.9).
Remark 4.4.We can establish in a similar way without major changes the results given above for the Dunkl intertwining operator and its dual in the multidimensional case.

3. 2 .Notation 3 . 1 .
Dunkl wavelets on R. Using the Dunkl transform Ᏺ D and the Dunkl translation operators τ x , x ∈ R, we define and study in this section Dunkl wavelets on R and the Dunkl continuous wavelet transform on R, and we prove for this transform Plancherel and inversion formulas.We have the following notations: .14) satisfies ∀y ∈ R, Ᏺ D α t (y) = e −ty 2 .(3.15)